In this paper, we build on the work of [T. Hughes, G. Sangalli, VARIATIONAL MULTISCALE ANALYSIS: THE FINE-SCALE GREENS' FUNCTION, PROJECTION, OPTIMIZATION, LOCALIZATION, AND STABILIZED METHODS, SIAM Journal of Numerical Analysis, 45(2), 2007] dealing with the explicit computation of the Fine-Scale Green's function. The original approach chooses a set of functionals associated with a projector to compute the Fine-Scale Green's function. The construction of these functionals, however, does not generalise to arbitrary projections, higher dimensions, or Spectral Element methods. We propose to generalise the construction of the required functionals by using dual functions. These dual functions can be directly derived from the chosen projector and are explicitly computable. We show how to find the dual functions for both the $L^2$ and the $H^1_0$ projections. We then go on to demonstrate that the Fine-Scale Green's functions constructed with the dual basis functions consistently reproduce the unresolved scales removed by the projector. The methodology is tested using one-dimensional Poisson and advection-diffusion problems, as well as a two-dimensional Poisson problem. We present the computed components of the Fine-Scale Green's function, and the Fine-Scale Green's function itself. These results show that the method works for arbitrary projections, in arbitrary dimensions. Moreover, the methodology can be applied to any Finite/Spectral Element or Isogeometric framework.

翻译：本文基于[T. Hughes, G. Sangalli, 变分多尺度分析: 细尺度格林函数、投影、优化、局部化及稳定方法, SIAM数值分析期刊, 45(2), 2007]中关于细尺度格林函数显式计算的工作展开。原始方法选择一组与投影算子关联的泛函来计算细尺度格林函数，然而这些泛函的构造无法推广至任意投影、高维情况或谱元方法。我们提出通过对偶函数来推广所需泛函的构造。这些对偶函数可直接从所选投影算子导出，且具有显式可计算性。我们展示了如何求得$L^2$投影和$H^1_0$投影的对偶函数，进而证明基于对偶基函数构造的细尺度格林函数能够一致地再现被投影算子移除的未解析尺度。该方法通过一维泊松问题、一维对流扩散问题以及二维泊松问题进行了验证。我们展示了细尺度格林函数的各计算分量及细尺度格林函数本身。结果表明该方法适用于任意投影和任意维度，且可应用于任意有限元/谱元或等几何分析框架。